If the sum and product of the mean and variance of a binomial distribution are $15$ and $54$ respectively,then the number of trials in it is

If $\mu$ and $\sigma^{2}$ are the mean and variance of a random variable $X$ whose probability mass function is given by $P(X=x) = \binom{6}{x} \left(\frac{1}{3}\right)^{x} \left(\frac{2}{3}\right)^{6-x}$ for $x = 0, 1, 2, \ldots, 6$,then the value of $2\mu + 12\sigma^{2}$ is:

If the mean and variance of a binomial variable $X$ are $2$ and $1$ respectively,then $P(X>1)=$

If $X \sim B(5, p)$ is a binomial variate such that $P(X=3)=P(X=4)$,then $P(|X-3| < 2)=$

If $A$ and $B$ throw two dice $100$ times each simultaneously,then the probability that both of them will get an even number as the total score at the same time in all the throws is

From a lot of $30$ bulbs which include $6$ defectives,a sample of $4$ bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.